Geoscience Reference
In-Depth Information
Once the α's and β's have been estimated (via maximum likelihood), the esti-
mated survival and capture probabilities for any animal can then be calcu-
lated using the logistic equations.
One problem with using logistic regression functions with modeling sur-
vival probabilities is that there is no simple way to take into account changes
in the time involved. For example, if there are 2 years between some samples
and only 1 year between others, then this cannot be allowed for in a simple
way by introducing into the model a parameter of time between samples.
A better function to use in this respect would be the proportional hazards
function ϕ
j
= exp{−exp(−
u
j
)
t
j
} where
t
j
is the time between when samples
j
and
j
+ 1 are taken. So far, this approach to modeling survival does not seem
to have been used. The approach usually taken is to model ϕ raised to the
power
t
j
as a logistic function of covariates, then report the
t
j
th root of this
parameter. This technique assumes survival of a period is a power function
of the base survival (survival of 1 time unit) raised to the interval length
(i.e., ϕ
j
t
). This model may or may not be valid.
Some care is needed in setting up models for which parameters vary with
time. For example, consider the logistic functions of Equations (8.22) and
(8.23) for survival and capture probabilities in the CJS model with
k
samples.
This model has 2k - 2 parameters ϕ
1
, ϕ
2
, . . . , ϕ
k
-1
and
p
2
,
p
3
, . . . ,
p
k
, but it is only
possible to estimate the product ϕ
k
-1
p
k
, rather than separate values for ϕ
k
-1
and
p
k
. One way to handle this within the logistic framework involves setting
ϕ
k
-1
= ϕ
k
-2
and using the parameterization
exp(
α+α+
)/{1 exp(
α+α
),
jk
jk k
1
≤
≤−
3
0
j
0
j
φ=
j
exp(
α+
)/{1 exp(
α
),
=
−
2 and
−
1
0
0
and
exp(
β
)/{1 exp(
+
β
),
j
=
2
0
0
p
=
j
exp(
)/{1 exp(
),
3
jk
β+β
+
β+β
≤
≤
0
j
0
j
The 2k - 3 parameters in this case are α
0
, α
1
, . . . , α
k
-3
and β
0
, β
3
, β
4
, . . . , β
k
,
which can all be estimated. Other alternative parameterizations would be
equally good in the sense that they would give the same estimates of ϕ
1
,
ϕ
2
, . . . , ϕ
k
-2
and
p
2
,
p
3
, . . . ,
p
k
-1
.
This type of complication occurs with other models as well. It is therefore
important to know which parameters can be estimated from the available
data and adopt an appropriate set of parameters for the logistic functions.
Unfortunately, the variable metric algorithm for maximizing the likelihood
function that is currently favored (see the following discussion) is capable of
determining this maximum even when the model being fitted contains more
